611 6.2 The Unit Circle and Circular Functions EXAMPLE 3 Approximating Circular Function Values Find a calculator approximation for each circular function value. (a) cos 1.85 (b) cos 0.5149 (c) cot 1.3209 (d) sec1-2.92342 SOLUTION (a) cos 1.85 ≈ -0.2756 Use a calculator in radian mode. (b) cos 0.5149 ≈0.8703 Use a calculator in radian mode. (c) As before, to find cotangent, secant, and cosecant function values, we must use the appropriate reciprocal functions. To find cot 1.3209, first find tan 1.3209 and then find the reciprocal. cot 1.3209 = 1 tan 1.3209 ≈0.2552 Tangent and cotangent are reciprocals. (d) sec1-2.92342 = 1 cos1-2.92342 ≈ -1.0243 Cosine and secant are reciprocals. S Now Try Exercises 35, 41, and 45. Radian mode This is how the TI-84 Plus calculator displays the results of Example 3, fixed to four decimal places. Determining a Number with a Given Circular Function Value We can reverse the process of Example 3 and use a calculator to determine an angle measure, given a trigonometric function value of the angle. Remember that the keys marked sin−1, cos−1, and tan−1 do not represent reciprocal functions. They enable us to find inverse function values. For reasons explained in a later chapter, the following statements are true. • For all x in 3-1, 14, a calculator in radian mode returns a single value in C - p 2 , p 2 D for sin-1 x . • For all x in 3-1, 14, a calculator in radian mode returns a single value in 30, p4 for cos-1 x . • For all real numbers x, a calculator in radian mode returns a single value in A - p 2 , p 2 B for tan-1 x . EXAMPLE 4 Finding Numbers Given Circular Function Values Find each value as specified. (a) Approximate the value of s in the interval C 0, p 2 D if cos s = 0.9685. (b) Find the exact value of s in the interval C p, 3p 2 D if tan s = 1. SOLUTION (a) Because we are given a cosine value and want to determine the real number in C 0, p 2 D that has this cosine value, we use the inverse cosine function of a calculator. With the calculator in radian mode, we find s as follows. s = cos-110.96852 ≈0.2517 See Figure 16. The screen indicates that the real number in C 0, p 2 D having cosine equal to 0.9685 is 0.2517. Radian mode Figure 16
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